if is a subring of , where has unity (let's call it ), and say is the unity of . Now, I have shown that it is NOT necessarily true that

Then if and are both fields (every element is a unit) then I'm trying to show that it is necessarily true that .

I am thinking you just take any , then , but since as well, . Is that legitimate reasoning? I'm not confident with my answer for some reason.

(2) Let for a commutative ring .

Suppose that is a zero divisor in . Prove that either or is a zero divisor.

So, Well, first what I did was use the defn of zero divisor (z.d.)
If is a z.d. then such that and

Then, just by associativity, so, well, can we assume that and ? Cause we'd be done - is a z.d. ! By commutativity, just switch and and then we're done.

Any help appreciated! Thanks!!

Jan 20th 2011, 01:41 PM

roninpro

Your proof for (1) looks okay. You just need to emphasize that inverses are unique.

You have a good idea for (2). To clean it up, you should note that neither nor should be zero, or else is zero, which is not what we want. Then, you are correct in taking nonzero so that and . So if , then you are done, since will satisfy the zero divisor definition. But what happens if ?

Jan 21st 2011, 04:38 AM

topspin1617

I'd just like to point out that (1) is actually true whenever the rings are integral domains.

In the ring , of course . Also, since and is the multiplicative identity of , we get .

Put these equalities together to get . Then, since we are in an integral domain, we can "cancel" the from both sides to conclude .